Too much or not enough crimes? On the ambiguous effects of repression

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1 MPRA Munich Personal RePEc Archive Too much or not enough crimes? On the ambiguous effects of reression Eric Langlais BETA, CNRS and Nancy University 12. January 2007 Online at htt://mra.ub.uni-muenchen.de/1575/ MPRA Paer No. 1575, osted 25. January 2007

2 Too much or not enough crimes? On the ambiguous effects of reression Eric Langlais Bureau d Economie Théorique et Aliquée, CNRS and University of Nancy January 15, 2007 Abstract Theuroseofthisaeristoinvestigate the otimal enforcement of the enal code when criminals invest in a secific class of avoidance activities termed dissembling activities (i.e. self-rotection efforts undertaken by criminals to hedge their illegal gains in case of detection and arrestation). We show that the enal law has two screening effects: it searates the oulation of otential criminals between those who commit the crime and those who do not, and in the former grou, between those who undertake dissembling efforts and those who do not. Then, we show that it is never otimal to use less than the maximal fine in contrast to what may occur with avoidance detection (i.e. efforts undertaken in order to reduce the robability of arrestation: Malik (1990)); and furthermore, that the otimal enal code may imly overdeterrence. Finally, we show that any reform of the enal code has ambiguous effects when criminals undertake dissembling activities which are a by-roduct of illegal activities, since increasing the maximum ossible fine may increase or decrease the number of crimes committed and may increase or decrease the roortion of illegal gains hedged by criminals. Keywords: deterrence, avoidance detection, dissembling activities, otimal enforcement of law. JEL Classification: D81, K42. Acknowledgements: We are thankful to Andreea Cosnita for her remarks. 1 Introduction The canonical economic literature on crime and unishment initiated by Becker (1968) has rovided two classical results. On the one hand, the best trade-off between robability and enalty is achieved when monetary enalties are set to their maximum ossible level, because fines are most of the time costless, allowing the enforcement authority to set them as high as ossible. On the other 1

3 hand, it is not otimal to comletely deter individuals from engaging in an illegal activity, since for at least some individuals, the gains from engaging in the roscribed activity may be sometimes larger than the external costs it imoses on the rest of the society. The first result has romted a large body of literature (see Garoua (1997) or Polinsky and Shavell (2000) for surveys) discussing cases where fines are costly resources for enforcers or for the criminals, hence justifying that less than maximum fines be used. In contrast, the second result is a common by-roduct of the former, and it has been shown that whenever enforcement authorities have imerfect information about criminals activities and/or their characteristics, the otimal design of the enal code allows some level of underdeterrence to exist. Following this line, we tackle in this note two commonly acknowledged results: on the one hand the fact that avoidance activities undertaken by criminals are a major reason justifying the otimality of less than maximum fine (Malik (1990), Bebchuk and Kalow (1993)), and on the other hand, that such activities aggravate the issue of criminals underdeterrence (Sanchirico (2006)). In contrast, we will rove first that for the secific class of avoidance activities that we term dissembling activities, it is never otimal to use less than maximum fines. Second, we will also show that ublic olicies designed to revent criminal behavior may lead to overdeterrence, in the sense that some offenders are deterred from engaging in the illegal activity although their rivate benefit is larger than the external cost they imose on the rest of the society. Avoidance activities encomass various exenditures engaged by criminals in order to reduce their exosure to the risk of unishment. It comrises installing radar detectors to avoid seeding tickets, lobbying oliticians to relax the enforcement of regulations, bribing an enforcement agent to let go free a culrit, destroying or covering u incriminating evidences, or investing in long and costly litigations and so on. Thus, we suggest a basic albeit more comrehensive tyology similar to the distinction made in the economics of insurance markets, between self-rotection and self-insurance. In fact, some avoidance activities are undertaken in order to lower the robability of arehension, conviction and/or unishment. Tyically, this is the case for examle with radar detectors. Note that such exenditures may be understood as self-rotection investments from the oint of view of criminals (they are more secifically termed avoidance detection by Sanchirico (2006)). But the rationale for others kind of avoidance activities is in contrast to reduce the imact of the arrestation and unishment on the wealth or welfare of the criminals: tyically, it occurs when criminals are strategically bankrut or non solvable, as it is the case when they render non seizable the benefits of the crime. In this case, it corresonds for the criminals to a kind of self-insurance behavior that will be termed dissembling activities in the aer. In fact, the existing literature on avoidance activities focuses on the case of detection avoidance. Sanchirico (2006) has recently suggested that it is a serious limit to the effectiveness of ublic olicies in the area of crime deterrence. He argues that it imlies the unfortunate but unavoidable result that any increase in ublic monitoring exenditures leads to an increase in criminals avoidance 2

4 activities, which in turn has an adverse feedback effect on the effectiveness and efficiency of ublic detection, thruly leading to a high level of underdeterrence. Nevertheless, Sanchirico does not address the issue of the otimal robability/fine trade-off. Such an analysis has been earlier rovided by Malik (1990) and Bebchuk and Kalow (1992) who have shown that avoidance detection may justify that less than maximum fines are otimal 1. Here, we focus on dissembling activities, assuming that criminals investments in order to avoid the risk of unishment enable them to hedge their illegal benefits in case of arrestation, allowing the enforcer to seize only a small amount of those outcomes. Section 2 describes the basic set u used in the aer, and roves that the enal code has two different screening effects: on the one hand, it searates the oulation of offenders between those who commit the crime and those who do not; on the other, it also distinguishes among the active criminals between those who undertake dissembling efforts and those who do not. In section 3, we show that the beckerian result, namely the otimality of maximum fines, still holds here. However, and in contrast to what occurs both in Becker s aer or in Malik s aer, overdeterrence may now occur at the otimum. Section 4 focuses on the effectiveness of ublic interventions. We first show that monetary enalties and the robability of control may be either substitutable or comlementary instruments. This imlies that when enforcement olicies become more reressive, criminals may take countervailing decisions which result in more crimes, more individuals making dissembling efforts and saving a larger roortion of their illegal benefits in case of arrestation. Finally, this means that the reform of the enal code has ambiguous effects on criminality: in the situation where underdeterrence exists at the otimum, the distortion from the first best may be reduced as the maximal level of fine increases (for examle, with the seizable wealth or assets of criminals) since the otimal level of deterrence goes closer to the external cost of crimes: ublic olicies become thus more efficient. On the contrary, in the case where overdeterrence occurs at otimum, then the distortion with resect to the first best may be aggravated as the maximal ossible fine is raised, making the level of deterrence closer to full deterrence. Section 5 briefly concludes. 2 Criminals behavior Let us consider the case where the illegal activity allows the (risk neutral) criminal to obtain a benefit equaltob (and b =0if the illegal act is not undertaken) which will be called the tye of the criminal. Public authorities do not observe the tye b. They just know that b is distributed according to a uniform distribution function on [0,B]. On the other hand, the (external) loss to the rest of the society is D<Bin case of crime, whatever the rivate benefit forthe criminal 2. We consider here that ublic enforcers are endowed with two basic 1 See also Nussim and Tabbach (2006). 2 Thus as usual in the literature, the first best level of deterrence corresonds to the illegal benefit b = D (assuming it can be obtained at a small enforcement cost). Given that the tye 3

5 instruments, as is usual in the literature: monetary sanctions (enalty or fine) f>0, and exenditures in the monitoring of criminals behavior, defined for the sake of simlicity as the choice of a robability of control (encomassing arrestation, conviction and unishment for an illegal behavior). When he is caught, the offender has to ay the fine but the rotective measures undertaken ex ante allow him to save only the fraction β(x) ]0, 1[ of his benefit b; x denotes the effort in the dissembling activity (caution) and we assume that β(0) = 0, β 0 > 0, β 00 < 0. Furthermore, we assume that the monetary equivalent of the disutility cost of criminal s efforts is simly v(x) =x.the maximum exected benefit obtained by the criminal when he undertakes the illegal activity and makes the avoidance effort is equal to: u max (π(x, )b f x) (1) x with π(x, ) =1 + β(x), which may be understood as the ex ante total roortion of the illegal benefitsavedbytheoffender 3. The individually otimal behavior of a criminal is described by the following roosition, denoting ˆx the efficient level of effort 4 : Proosition 1 The oulation of criminals searates in three different grous, defined according to two different thresholds of the benefits labelled b and b, such that: i) if the criminal s tye is b [0, b], then he does not commit the crime and makes no dissembling efforts (ˆx =0); ii) if the criminal s tye is b ] b, b ], then he does commit the crime but without undertaking any dissembling effort (ˆx =0); iii) if the criminal s tye is b ]b,b], then he does commit the crime and undertakes a ositive level of effort (ˆx >0) which satisfies: β 0 (ˆx)b =1 (2) Proosition 1 means that any enforcement olicy has in fact two distinct screening effects on the oulation of otential criminals. On the one hand, it leeds to the searation between those who become active criminals, and those who are deterred - this a basic effect. The threshold b 1 f corresonds to the level of deterrence under which no crime is committed (this threshold increases both with f and ). But there exists a second effect: among the active offenders, some of them will also invest in dissembling activities (make some efforts to hedge their benefits in case of arrestation), while the others will not. Namely, b 1 β 0 (0) is the threshold over which any crime committed is accomanied by an effort in dissembling activities (and it decreases with but of the criminals is not observable, it is generally never attainable. 3 With robability 1, the criminal saves the benefit b in roortion 1, although with robability he saves only β(x) < 1. 4 All the roofs are in the aendix. 4

6 is indeendant of f). It is easy to see 5 that for any b ]b,b], theotimalˆx is unambiguously increasing with and b, but is indeendent from the fine. The value of the fine f matters only in the sense that it influences the decision to engage in the illegal activity or not, although it does not affect the decision to undertake or not the avoidance exenditures. The rest of the aer studies the effects of the otimal enforcement of the law on this secific structure of the oulation of criminals. 3 Second best olicies The management costs associated with the monetary enalty are neglectable, but monitoring the criminal activity entails a cost equal to m(), withm 0 > 0 and m 00 > 0. The government has to choose a fine f and a robability of control in order to maximize the social welfare function: S = 1 B Z b b ((1 )b D)db + 1 B Z B b (π(x(, b),)b x(, b) D)db m() under the constraint 6 f F.Thetwofirst (integral) terms in S corresond to the exected rivate benefit associated with the illegal activity (the benefit of the criminal without dissembling efforts minus the external cost, lus his benefit when he commits the crime with an ositive effort minus the cost of dissembling and the cost to the society). The last one is the cost of monitoring for ublic authorities. The fine is a mere transfer between the (risk neutral) criminal and the government, it does not aear in the social welfare function (it is not worth from a social oint of view). It is obvious (see also Malik (1990)) that for small values of the external cost of crime and/or large values of the ublic cost of monitoring, the solution of this roblem may be zero deterrence; and under the oosite conditions (large values of the external cost of crime and/or small values of the ublic cost of monitoring), we may obtain comlete deterrence. Thus, we focus rather on the more owerful case with conditional deterrence hereafter. If an interior solution (, f) exists, it satisfies the first order conditions of maximization which are written: " µ 1 b (D f) B (1 ) = m0 + 1 x β b B β 0 (0) + ˆx b = Z b 5 Alying the imlicit function theorem to (2), one obtains: β0 (ˆx) β 00 (ˆx)b > 0. b bdb + Z B b ˆx = β0 (ˆx) β 00 (ˆx) # (1 ˆβ)bdb (3) > 0 and 6 This is the most natural secification when we consider that the cost of avoidance corresonds to the disutility of criminals efforts, and F corresonds to the seizable wealth or assets of criminals. 5

7 1 (D f) B 1 = λ (4) with λ =0if f<fbut λ>0 otherwise, and denoting β = β(x ), ˆβ = β(ˆx). More secifically, the LHS in (3) is the social marginal benefit from the control of illegal activities, while the RHS corresonds to the social marginal cost of controling which takes into account the enforcer s marginal cost of monitoring (first term) and the criminals marginal cost of dissembling effort (last three terms). Similarly, the LHS in (4) is the social marginal benefit offines, and the the RHS is their social maginal cost (which is simly the shadow rice of the constraint, since fines are costless). In the aendix, we rove that the following results hold: Proosition 2 The solution with conditional deterrence has the following roerties: i) The maximum fine f = F is always otimal, and the robability must be set as small as ossible according to (3). ii) We obtain that F < D and there may exist either over or underdeterrence at otimum ( b 1 F D). Result i) is in contrast to the one obtained by Malik (1990) in the case of detection avoidance i.e. when avoidance activities enable criminals to lower the robability of arrestation and unishment: whereas less than maximum fine may be otimal under detection avoidance, this never occurs under dissembling activities. These two different results are easily exlained. Under dissembling activities, criminals effort are indeendent of the fine: raising the fine entails no additional costs on criminals (beyond the exected fine aid in case of arrestation), and thus has only the direct effect on deterrence. Hence, insufficient deterrence obtaines unless maximum fines are set. In contrast, with detection avoidance (Malik (1990)), the fines imose a rivate cost on criminals, over the exected fine aid in case of arrestation; deending on whether avoidance exenditures become more or less sensitive to the fine, then the enforcement autorities may use less than maximum fine or not. Part ii) also challenges the usual result of the literature. In the canonical model of Becker, there is not enough deterrence at the otimum: some of the criminals for which the benefit of committing the crime is smaller than the external cost on the society, are not deterred. This is exlained by the fact that the level of deterrence corresonds to the exected fine aid by criminals when they are arrested - and random detection is justified by the costly resources used to control criminal activities. In contrast, in the resent set u the exected fine is always smaller than the external cost of crime but does not determine the level of deterrence: this latter is set at a threshold high enough to deter only those in the oulation of criminals who would never engage in dissembling activities; but on the other hand, as far as it is socially worth to deter also some 6

8 of the active criminals who do make an effort (as soon as their benefit islower than D), it may be necessary to set the robability of control at a high level given that these individuals are not sensitive to the level of the fine. As a result, the robability may be sometimes set at a level high enough to induce excessive deterrence at otimum. But deending on the roerties both of the disutility cost of effort and of the technology of dissembling (see the three last terms in the RHS of (3)), the oosite result of underdeterrence may arise, as usually found both by Becker (1968), as well as Malik (1990) or Sanchirico (2006) under detection avoidance. Note that it could be ossible also that the first best level of crimes occurs (by chance). Nevertheless, in such a case, due to the asymmetric information the enal code always imoses an excessive cost to the society: among the criminals who are not deterred, some do make a dissembling effort although their activity is valuable (i.e. they would never be unished if their tye were observable), and the distulity cost of their effort reduces the social welfare. Finally, roosition 2 imlies that although maximum fines are always otimal, they have an ambiguous effect on the number of crimes committed when criminals tye is not observable, and when some of them invest in dissembling activities. In the following, we investigate the consequences of this result more deely. 4 Countervailing behaviors of reressive olicies As roven by Garoua (2001), although the canonical result of Becker is usually understood as establishing the substituability between both instruments, this is not necessarily true. Let us focus here on the degree of substitutability/comlementarity between fines and controls, i.e. whether the otimal robability decreases or increases with the maximum fineintheresenceofdissem- bling activities. Alying the imlicit function theorem to (3) with f = F,itis easy to verify that sign d df = signs F with: S F = 1 D 2F B (1 ) 2 (5) Hence the following result is straightforward: Proosition 3 When the fine increases, the otimal robability of control may either decrease or increase. This is essentially the same result as the one obtained in the canonical model without avoidance activity; see Garoua (2001) for a more detailed discussion of its intuitive meaning: when the maximal fine is high, the level of deterrence is also high (and there may exist overdeterrence); thus, raising F, the enforcer has the oortunity to decrease the robability in order to reduce the enforcements 7

9 costs. In contrast, when the maximal fine is small (to the limit, close to zero), the level of deterrence is also small, and it may be worth in this case to raise both F and the robability in order to reach enough deterrence. This first finding has several imlications. The next roosition focuses on the imact of the monetary sanctions on the distorsion to the firstbestnumberof crimes, on the number of criminals undertaking an effort and on the roortion of illegal benefits they can save in case of detection. Proosition 4 An increase in the maximum fine yields: i) an increase in the level of deterrence b when the robability and the fine are comlements, but an ambiguous effect when they are substitutes. ii) a decrease in the number of active criminals undertaking dissembling efforts b b, if the robability and the fine are comlements, but an ambiguous effect if they are substitutes. The ambiguity in art i) of roosition 4 is easily exlained by the fact that an increase in F has a direct effect on b which is always ositive, but also an indirect effect through the variation of which is ositive when and F are suosed to be comlementary, but negative when they are substitutable: thus, the total effect deends on whether the first or the second one dominates. The indirect effect also holds in art ii) since the threshold b does not directly deend on F but is sensitive only to the fequency of control. An obvious imlication of roosition 4 is that it is very uncertain whether the enforcement authority has the oortunity to reach a fine tuning of the level of criminality when criminals invest in dissembling activities. In fact, the following results generally hold: Corollary 5 An increase in the maximum fine: i) always yields a decrease (resectively, an increase) in the level of underdeterrence (overdeterrence) when the robability and the fine are comlements at the otimum, whereas the effect is ambiguous when they are substitutes. ii) has no effect on the benefits saved by the marginal criminal b. iii) imlies a decrease (an increase) in the benefits saved by any criminal b> b when the robability and the fine are substitutes (resectively comlements). Notice first that the ambiguity in i) arises only when the robability and the fine are substitutes. Secondly, when the robability and the fine are comlements, increasing the sanction has favorable effects in case of underdeterrence, but adverse ones in case of overdeterrence. When underdeterrence occurs at the otimum, then the otimal level of deterrence goes closer to the external cost of crimes as the maximum fine grows u; in other words, the distortion to the first best level of deterrence is reduced, and ublic olicies become more efficient. On the contrary, when overdeterrence occurs at the otimum, then 8

10 the distortion with resect to the first best increases as the maximum fine is raised, making the level of deterrence closer to full deterrence. Concerning ii) and iii), the results reflect that the roortion ˆβ for any b b do not directly deend on F but is sensitive only to the fequency of control. Thus for art ii), the increase in the fine has two indirect but oosite effects on the marginal criminal b through its dissembling efforts: one on the robability of control, the other on the illegal benefit b, but given that they have the same magnitude, the net effect is nul. In contrast, in iii) for any b>b the effect is simly the one associated to the robability of control. 5 Final remarks This aer rovides a different view on the effects of the enal code when criminals have the oortunity to undertake avoidance activities. We have modified Malik (1990) s model to incororate a continuum of criminals and we assume that those criminals have the oortunity to invest in dissembling activities which allow them to hedge the benefits of the crime when they are arrested and unished (revent that illegal assets be seized by the enforcer). In this set u, we show that the adotion by criminals of such self-rotective measures has major consequences: secifically, we show that maximum fines are always otimal, and that overdeterrence may be otimal. This differs from the results reviously obtained by Malik (1990) or Sanchirico (2006): avoidance activities are usually exected to justify the use of less than maximum fines, and to aggravate the roblem of underdeterrence which initially aeared in the canonical world àla Becker. More generally, it also challenges the common view which is to condition the design of law enforcement on the seizable wealth of criminals since the maximal ossible fine is commonly interreted as the individual wealth of criminals. For examle, Garoua (2001) concluded that the otimal robability is an inversed U-shaed function in criminals wealth (both small and large criminals face a low robability of sanction) when wealth is a ublic information (observable before detection and rosecution). In contrast, when wealth is a rivate information (Polinsky and Shavell (1991)), the otimal robability is U-shaed with resect to criminal s wealth (both small and large criminals face a large robability of sanction). Our results suggest that in the resence of dissembling exenditures, which are an unavoidable by-roduct of illegal activities, things are less clear, and more restrictive olicies may have counterintuitive and/or adverse effects. References Bebchuk A. and Kalow L. (1993), Otimal sanctions and differences in individuals likelihood of avoiding detection, International Review of Law and Economics, 13, Becker G. (1968), Crime and unishment: an economic aroach, Journal of Political Economy, 76,

11 Garoua N. (1997), The theory of otimal law enforcement, Journal of Economic Surveys, 11, Garoua N. (2001), Otimal magnitude and robability of fines, Euroean Economic Review, 45, Malik A. (1990), Avoidance, screening and otimum enforcement, RAND Journal of Economics, 21, Nussim J. and Tabbach A. (2006), Punishment, deterrence and avoidance, mimeo, Tel Aviv University. Polinsky M. and Shavell S. (1991), A note on otimal fines when wealth varies among individuals, American Economic Review, 81, Polinsky M. and Shavell S. (2000), The economic theory of ublic enforcement of law, Journal of Economic Literature, 38, Sanchirico C. (2006), Detection avoidance, New York University Law Review, 81, Stigler G. (1970), The otimum enforcement of laws, Journal of Political Economy, 78, APPENDIX Proofofroosition1 Remark firstthatthereexistatleastsomevaluesofb ]0,B] such that: β 0 (0)b 1 0 b b 1 β 0 (0): then it is individually efficient for such criminals to make no dissembling efforts otherwise their exected benefit would be decreasing with x. Thus, for any criminal of tye b b there are two ossibilities: A/ either the condition (1 )b f 0 b b 1 f also holds and the criminal does not enter into the illegal activity; this roves i); B/ or b> b, andthecriminaloftyeb b commits the crime but chooses a ˆx =0.Thisrovesii). On the other hand, for any b>b if an interior solution ˆx >0 exists, it must satisfy (2) which is necessary and sufficient given the assumtions ut on β. Remark now that the associated u in (1) writes u =[(1 )b f]+bβ(ˆx) ˆx: given that the first bracketed term is ositive as soon as b> b, while the second one is also ositive for any b>b since β(x) is increasing and concave, then it is obvious that any criminal b>b engages in the illegal activity and makes a ositive effort in the dissembling activity. This roves iii). Proofofroosition2 i) Let us consider a solution where the otimal fine satisfies f<f.accordingto(4),thisimliesthatf = D and thus for any ositive robability we obtain 10

12 S = m 0 1 B " µ x β β 0 (0) b + Z b b bdb + Z B b # (1 ˆβ)bdb < 0 since according to the concavity roerty: x β β 0 (0) > 0, whichisacontradiction to the assumtion that it is otimal to control. As a result f = F is otimal, and must be set as low as ossible according to the condition (3). It is easy to verify that the second order condition is satisfied as long as m has enough decreasing returns to scale (left to the reader). ii) Finally, given that the RHS of (4) is ositive, it must be that D>F. Hence, there may exist either over or underdeterrence at otimum, given that b 1 F D. Proofofroosition4 It is straightforward to show that: d b df = 1 F S F (1 ) 2 S where d df = S F S, with S F > 0 when and F are comlements, but S F < 0 when and F are substitutes. S is negative (by the second order condition) but has several terms either ositive or negative since the rivate marginal cost of effort in (3) is not monotonic in (left to the reader), such that FS F (1 )S 0. Hence the result i). The imact on the difference b b is: d df (b b) = 1 which is negative if d df result ii). µ 1 2 β 0 (0) + Proofofcorollary5 At the threshold b we have: where e x x ii). But for any b>b,wehave: F d (1 ) 2 df > 0 but has an ambiguous sign otherwise. This is dβ df = 0 x d β df (ex e x b ) x = β0 (x ) β 00 (x )x = e x b x b b x : hence dβ df =0.Thisroves dˆβ df = ˆβ 0 ˆx d eˆx df which has the sign of d df. Hence the result iii). 11